# How to convert multi-band Energy Time Curve (ETC) to Impulse Response (IR)?

I am running my own sound simulator. Now I have obtained the Energy Time Curve (ETC, expressed by energy) by physical simulation, and they are stored in 8 discrete frequency bands (because decay of sound in the air is dependent on sound frequency). Now I am having problem getting the impulse response (IR, expressed by amplitude) of the measured system. I will need the IR to convolve with some audio files.

So I have $ETC(f,t)$, where $f\in\{f_1,f_2,...,f_8\},t=1,2,...N$. And the length $N$ is sufficient to capture the energy decay under a sample rate of 48kHz. How should I derive $IR(t)$ from the above $ETC$?

• i think you need to be more explicit about what you mean by the ETC. i remember the Heyser days and the ETC was the square of the envelope of what EEs call the analytic signal. is it that. something like $\mathrm{ETC}(f,t)$ looks to me to be something like the magnitude square of the STFT. with the latter, you lose phase information and an All-Pass Filter and a delay line and even a wire will look the same (but they have different IR). – robert bristow-johnson Jul 21 '17 at 5:56
• I know of people who have good results using RAM PE in air acoustics staff.washington.edu/dushaw/AcousticsCode/…. I recall that there is an example of propagating a DFT of a broadband signal and reconstructing it. I'm not familiar with the term ETC so I'm guessing that is similar to your problem – user28715 Jul 23 '17 at 4:53
• Does your ETC have a form of echograms? If it does then you can synthesize the IR from band-pass echograms. This paper contains a very simple method to do that. – jojek Jul 4 '20 at 15:15